Answer:
So, Katelyn deposited $2500 in a CD, and the interest rate is 4.25% compounded daily. That means her money will grow every day at a rate of 4.25%/365 (since there are 365 days in a year).
To find out how many years it will take Katelyn to earn $250 in interest, we need to use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A is the amount of money in the account after t years
P is the principal amount (the initial deposit)
r is the annual interest rate (as a decimal)
n is the number of times the interest is compounded per year
t is the time in years
In this case, we know:
P = $2500 (the initial deposit)
r= 0.0425 (4.25% as a decimal)
n = 365 (compounded daily)
t is what we're trying to find
We also know that Katelyn wants to earn $250 in interest, so her total balance after t years should be $2500 + $250 = $2750.
Plugging all of this into the formula, we get:
$2750 = $2500(1 + 0.0425/365)^(365t)
Simplifying a bit, we get:
1.01 = (1 + 0.0425/365)^(365t)
Taking the natural log of both sides, we get:
ln(1.01) = ln(1 + 0.0425/365)^(365t)
Using the property of logarithms that ln(a^b) = b*ln(a), we can simplify further:
ln(1.01) = 365t * ln(1 + 0.0425/365)
Finally, we can solve for t:
t = ln(1.01) / (365 * ln(1 + 0.0425/365))
Plugging this into a calculator, we get:
t ≈ 4.69 years
So it will take Katelyn about 4.69 years to earn $250 in interest on her CD.
I hope that helps!